Electromagnetic Casimir effect and the spacetime index of refraction

Abstract

In a recent paper [1] using a conjecture it is shown how one can calculate the effect of a weak stationary gravitational field on vacuum energy in the context of Casimir effect in an external gravitational field treated in 1 + 3 formulation of spacetime decomposition.. In this article, employing quantum field theory in curved spacetime, we explicitly calculate the effect of a weak static gravitational field on virtual massless scalar particles in a Casimir apparatus. It is shown that, as expected from the proposed conjecture, both the frequency and renormalized energy of the virtual scalar field are affected by the gravitational field through its index of refraction. This could be taken as a strong evidence in favour of the proposed conjecture. Generalizations to weak stationary spacetimes and virtual photons are also discussed. * Electronic address: bornadel@khayam.ut.ac.ir † Electronic address: nouri@theory.ipm.ac.ir, corresponding author 1

Casimir effect of two conducting parallel plates in a general weak gravitational field

“…However, in [15][16][17][18][19], the computations have been done through a lengthy and cumbersome method of the Green functions. In a paper by the author [24], it is shown that the energy density (i.e. the 00 component of the energy-momentum tensor) of the electromagnetic field in Fermi spacetime is exactly the same as the energy density of the two scalar fields mentioned in [24].…”

“…We know also that both ψ and φ satisfy the Klein-Gordon equation separately (see the appendix in [24]). The boundary condition for the electric field on the plates is E ⊥ (z = 0) = E z (z = 0) = 0 and E ⊥ (z = l) = E z (z = l) = 0, which in turn can be recast into boundary conditions on ψ and φ according to Eq.…”

“…In a paper by the author [24], it is shown that the energy density (i.e. the 00 component of the energy-momentum tensor) of the electromagnetic field in Fermi spacetime is exactly the same as the energy density of the two scalar fields mentioned in [24]. The method was used without any reference to the Green function method frequently used in the literature.…”

“…(33). In [24] it has been shown that boundary conditions for the electric and magnetic fields return a Dirichlet boundary condition on φ and a Neumann boundary condition on ψ. We have shown in the previous section that both of these conditions will lead to the same frequency shift.…”

“…In fact, some non-trivial topologies in curved spacetime do the same job as a boundary does [6,7]. The Casimir energy in curved spacetime has also been analyzed by many authors ( [1,2,6,[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and references therein). Recently, a Casimir apparatus consisting of two ideal conducting parallel plates in the weak field limit of the Kerr and the Horava-Lifshitz spacetimes has been studied in [8][9][10][11].…”